Gustafsson, I. (1979) Stability and Rate of Convergence of Modified Incomplete Cholesky Factorization Method. Research Report 79.02R, Department of Computer Sciences, Chalmers University of Technology and University of Göteborg, Göteborg, Sweden.

has been cited by the following article:

• TITLE: A Simple Proof of Gustafsson’s Conjecture in Case of Poisson Equation on Rectangular Domains

  AUTHORS: Gangjoon Yoon, Chohong Min

  KEYWORDS: Five-Point Finite Difference Method, Modified ILU Preconditioning, Condition Number

  JOURNAL NAME: American Journal of Computational Mathematics, Vol.5 No.2, May 20, 2015

  ABSTRACT: We consider the standard five-point finite difference method for solving the Poisson equation with the Dirichlet boundary condition. Its associated matrix is a typical ill-conditioned matrix whose size of the condition number is as big as . Among ILU, SGS, modified ILU (MILU) and other ILU-type preconditioners, Gustafson shows that only MILU achieves an enhancement of the condition number in different order as . His seminal work, however, is not for the MILU but for a perturbed version of MILU and he observes that without the perurbation, it seems to reach the same result in practice. In this work, we give a simple proof of Gustafsson's conjecture on the unnecessity of perturbation in case of Poisson equation on rectangular domains. Using the Cuthill-Mckee ordering, we simplify the recursive equation in two dimensional grid nodes into a recursive one in the level that is one-dimensional. Due to the simplification, our proof is easy to follow and very short.